# In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example:(i) If x ∈ A and A ∈ B, then x ∈ B(ii) If A ⊂ B and B ∈ C, then A ∈ C(iii) If A ⊂ B and B ⊂ C, then A ⊂ C(iv) If A ⊄ B and B ⊄ C, then A ⊄ C(v) If x ∈ A and A ⊄ B, then x ∈ B(vi) If A ⊂ B and x ∉ B, then x ∉ A

(i) Let us assume A = {1, 2}

And, B = {1, {1, 2}, {3}}

So,

And,

But,

Hence, the given statement is false

(ii) Let us assume, A {2}

B = {0, 2}

And, C = {1, {0, 2}, 3}

It is given in the question that,

But,

Hence, the given statement is false

(iii) It is given in the question that,

Let us assume,

So,

And,

Hence, the given statement is correct

(iv) It is given in the question that,

A B

And, B C

Let us now assume, A = {1, 2}

B = {0, 6, 8}

And, C = {0, 1, 2, 6, 9}

Hence, the given statement is false

(v) It is given in the question that,

And, A B

Let us now assume, A = {3, 5, 7}

And, B = {3, 4, 6}

As,

And,

Hence, the given statement is false

(vi) It is given in the question that,

A B

And, x B

Let us suppose, then we have:

But it is given in the question that, x B

x A

Hence, the given statement is true

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